Manuscript Em06-B@1 · Em06-B-v1.tex · lemma · 61e38859

Lemma: Sign-sensitive identities

Independent classical audit and compact-test estimates · lines 342–375 · proof included or linked

Em06-B@1#lem:identities, Em06-B@1#eq:Q-expanded, Em06-B@1#eq:CK, Em06-B@1#eq:QK

The following identities hold:

Q_{h} C (K ξ) Q_{K ξ} = - h^{μν} \partial_{μ} \partial_{ν} - C (h)^{ν} \partial_{ν} + \frac{1}{2} τ (h) P_{m}, = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .

For real $h$ , $Q_{h}$ is formally self-adjoint.

Proof. Expanding the divergence in (eq:Q) and using

\partial_{μ} a^{μν} (h) = - C (h)^{ν}

gives (eq:Q-expanded). The trace of $K ξ$ is $2 \partial_{μ} ξ^{μ}$ , and direct differentiation gives (eq:CK). Finally,

[P_{m}, X_{ξ}] = (□ ξ^{ν}) \partial_{ν} + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} .

Substituting $h = K ξ$ into (eq:Q-expanded) proves (eq:QK). Formal self-adjointness follows by integration by parts in the divergence form, including its real zeroth-order term.

Displayed formulas in this passage

Em06-B@1#eq:Q-expanded · line 345
$Q_{h} C (K ξ) Q_{K ξ} = - h^{μν} \partial_{μ} \partial_{ν} - C (h)^{ν} \partial_{ν} + \frac{1}{2} τ (h) P_{m}, = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .$
Em06-B@1#eq:CK · line 345
$Q_{h} C (K ξ) Q_{K ξ} = - h^{μν} \partial_{μ} \partial_{ν} - C (h)^{ν} \partial_{ν} + \frac{1}{2} τ (h) P_{m}, = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .$
Em06-B@1#eq:QK · line 345
$Q_{h} C (K ξ) Q_{K ξ} = - h^{μν} \partial_{μ} \partial_{ν} - C (h)^{ν} \partial_{ν} + \frac{1}{2} τ (h) P_{m}, = □ ξ, = (\partial_{μ} ξ^{μ}) P_{m} - [P_{m}, X_{ξ}] .$
unlabeled · line 361
$\partial_{μ} a^{μν} (h) = - C (h)^{ν}$
unlabeled · line 367
$[P_{m}, X_{ξ}] = (□ ξ^{ν}) \partial_{ν} + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} .$

Lemma: Sign-sensitive identities

Displayed formulas in this passage

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